Geodesic
Analysis of eigenvalues for problem on the transverse shear crack in material with power defining law
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 115 (2007) no. 2, pp. 60-68 Cet article a éte moissonné depuis la source Math-Net.Ru

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We research the non-linear eigenvalues problem which appears from the problem of determining the stress-strain state near the transverse shear crack tip in material with power relationship between strains and stresses. For the eigenvalues finding we use the perturbation method based on expanding the eigenvalue, corresponding eigenfunction and the index of non-linearity of material in series in terms of powers of small parameter, which is the difference between the eigenvalues related to the linear and non-linear problems. The comparison between the found eigenvalues and the exact numerical solution of considered non-linear eigenvalues problem is given. 
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L. V. Stepanova; M. E. Fedina. Analysis of eigenvalues for problem on the transverse shear crack in material with power defining law. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 115 (2007) no. 2, pp. 60-68. http://geodesic.mathdoc.fr/item/VSGTU_2007_115_2_a8/

[1] Hutchinson J. W., “Singular behavior at the end of tensile crack in a hardening material”, J. Mech. Phys. Solids, 16 (1968), 13–31 | DOI | Zbl

[2] Rice J. R., Rosengren G. F., “Plane strain deformation near a crack tip in a power-law hardening material”, J. Mech. Phys. Solids, 16 (1968), 1–12 | DOI | Zbl

[3] Yuan F. G., Yang S., “Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear”, Int. J. of Fracture, 69 (1994), 1–26 | DOI

[4] Nikishkov G. P., “An algorithm and a computer program for the three-term asymptotic expansion of elastic-plastic crack tip stress and displacement fields”, Engineering Fracture Mech., 50:1 (1995), 65–83 | DOI

[5] B. N. Nguyen, P. R. Onck, E. Van Der Giessen, “On higher-order crack-tip fields in creeping solids”, Trans. of the ASME. J. Appl. Mech., 67:2 (2000), 372–382 | DOI | Zbl

[6] Hui C. Y., Ruina A., “Why K? High order singularities and small scale yielding”, Int. J. of Fracture, 72 (1995), 97–120

[7] Williams M. L., “On the stress distribution at the base of a stationary crack”, Trans. ASME. J. Appl. Mech., 24 (1957), 109–114 | MR | Zbl

[8] Williams M. L., “Stress singularities resulting from varios boundary conditions in angular corners of plates in tension”, J. of Appl. Mech., 19 (1952), 526–528

[9] Meng L., Lee S. B., “Eigenspectra and orders of singularity at a crack tip for a power-law creeping medium”, Int. J. of Fracture, 92 (1998), 55–70 | DOI

[10] Chen D. H., Ushijima K., “Plastic stress singularity near the tip of a $\mathrm{V}$-notch”, Int. J. of Fracture, 106 (2000), 117–134 | DOI

[11] Neuber H., “Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law”, J. of Appl. Mech., 28 (1961), 544–550 | DOI | MR | Zbl

[12] Anheuser M., Gross D., “Higher order fields at crack and notch tips in power-law materials under longitudinal shear”, Arch. of Appl. Mech., 64 (1994), 509–518 | DOI | Zbl

[13] Naife A. Kh., Vvedenie v metody vozmuschenii, Mir, M., 1984 | MR

[14] Stepanova L. V., Matematicheskie metody mekhaniki razrusheniya, Sam. un-t, Samara, 2006

[15] Beiker Dzh., Greivs-Morris P., Approksimatsii Pade, Mir, M., 1986 | MR

[16] Kalitkin N. N., Chislennye metody, Nauka, M., 1978 | MR